In [1]} Davis and in [2]} Douglas derived the following fundamental overdetermined system
on the symmetric nondegenerate matrix \(\phi _{ij}=\frac{\partial ^2 L}{\partial \dot{y}^i\partial \dot{y}^j}=\phi _{ji}\) :
\(\frac{\partial \phi _{ik}}{\partial \dot{y}^j}=\frac{\partial \phi _{jk}}{\partial \dot{y}^i},\\\frac{d}{dx}\phi _{ij}+\frac{1}{2}\frac{\partial f^k}{\partial \dot{y}^i}\phi _{kj}+\frac{1}{2}\frac{\partial f^k}{\partial \dot{y}^j}\phi _{ki}=0,\\A_i^k\phi _{kj}=A_j^k\phi _{ki},\) 
